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Optimization Laboratory. Department of Operations Research. Stanford University. Stanford. California. USA- Dantzig, G.B. and P.W. Glynn. "Parallel Processon for Planning Under Uncertainty", Annals of Operation Research 22 (1990). pp. 1-2 1. Duncan. R. "A Survey of Parallel Computer Archi tecms". IEEE Trans. Cornputen ( 1990). Entnken. R. "Parallei Decomposition: Results for Staircase Linear Rograms". SIAM hnd of Optirniration, 1 ( 1996). pp. 96 1-977. November. Flynn. M.J. "Very High Speed Computing Systems". Proc. IEEE. 54 (1966), pp. 1901-1909. Fragniere. E.. J. Gondzio. R. Sarkissim. and J. P. Vial. "Structure Exploiting Tool in Algebnic Modeiing Lan pages". Management Science 16, No. 8 (2000). pp. 1 145- 1 1 58. Fngniere. E.. J. Gondzio. and J. P. Vid. "A Planning Mode! with one Million Scenvios Solved on an Affordable Panllel Machine". Logilab Technical Report (1998b), Section of Management Studies. University of Geneva. Geneva. Switzerland. GAMS Development Corporation. The GMS VO Library. 1996 Geist. A.. A. Beguelin. I. Dongam. W. Jiang. R. Manchek and V. Sunderam. 1994. A PVM: Panllel Vinual Machine - "Users' Guide and Tutorial for Networked Parailel Computing", The MIT Ress, Cambridge. MA. USA. Gnanendnn. S.K. and J.K. Ho. "Load Balancing in the Parallel Opirnization of Block- Angualu Linear Pro+ms". Mathematical Programrning 62 (1993), pp.4L-67. Ho. J.K.. "Conversence Behavior of Decomposition Algorithrns for Linear Rograms". Operations Research Leners 3 ( 1984). pp. 9 1-94. Ho. 3.K.. TC. Lee and R.P. Sundainj, 1988. "Decomposition of Linear Programs Using Parallel
Computation". Mathematical Programming 42 (1988). pp. 391-405. Ho. J.K. and E. Loute. 1996, "'On the de-gee of Decentralization in Linear Rogramming". Infownatica 7. pp. 337-348. Holmberg. K. "On the Convergence of Cross Decomposition". Mathemarical Progrming. 17 ( IWO). pp. 269-196. Holmberg. K. "'Linear Mem Value Cross Decomposition: A Generaiization of the Kornai-Liptak Method". Eu ropean Journal of Operarionol Research. 62 ( 1992). pp. 55-73. Konanek. K. 0. and Ji Shan Zhu. 1988. "New purification algorithms for linear pro_mamrning9*. Naval Research. Logistics. 35, No. 1. pp. 57 1-5 83. Lm. B. 1993. "A Prirnal-Dual Decomposition Method for Multi-Stage Linev Programs". Ph.D dissertation, Department of Management Sciences.Univenity of Waterloo. Waterloo. ON. CANADA. Lan. B. and J.D. Fuller. 1995a. "A hmal-Dual Decomposition Method for Two-Stage Lineu Ro_~s". working paper. Depment of Management Sciences. University of waterloo. Waterloo. ON. CANADA. Lm. B. and J.D. Fuller. 1995b. "A Pnmal-Duai Nested Decomposi tion Algorithm for Multi- Stage Lmear Pro_prns". worhng paper. Depment of Management Sciences. University of Waterloo. Waterloo. ON. CANADA. Murphy. F. K.. and M. V. Mudngeda. "A Decomposition Appmach for a Class of Economic Equilibrium Models ". Operations Research. M. No. 3 (1998). pp. 368-377. Sielsen. S. S .. and S. A. Zenios. "Scdable pdlel Benden decomposition for sta'hastic linear pro-orunming". ParaIIel Compu~ing, 23 ( 1997), pp. 1069- 1088. 159
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A PARALLEL PRIMAL-DUAL DECOMPOSITIO
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The University of Waterloo requires
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WATPAR, and in each of the tests, t
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To Soyoung and Katherine Eugene Par
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3.5 Sumary and Observations on the
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Table 4 . I Table 5.1 Table 5.2 Tab
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Chapter 1 Introduction 1.1 Brief Hi
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1.2 Motivation and Objectives of th
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2. Pmve and dernonstrate the conver
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Chapter 2 Literature Review This ch
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Benders komposition Method In the B
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in this algorithm, both subproblems
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SIMD vs MIMD Panllel computers cm b
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apan. In mmy LAN's, communication i
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angular linear programs by fixing t
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Chapter 3 Paralle1 Decomposition of
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ound constraints. The two parts are
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constnicted in the sarne way by res
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1T 17 t-IT T nlk*'T )' and &'-' = (
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a proposal is passed back to the fi
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END - solve ; if it is infeasible o
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l, and Theorem 3. lb mles out the p
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nonlinking constraints and upper bo
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(The "only if* part) If P is infeas
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. - Thmm 3.6 Suppose (-ri. )Jik, VI
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the optimal value has ken reached.
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spfl, we have have k .J< k PI -1 -
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zi - $' 5 z+ - &J-[, and by Theorem
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Chapter 4 Paraiiel Decomposition of
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upper bound constraints and nonnega
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type subproblem. The lower bound su
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An algorithm could be defined to ex
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Figure 43 9-stage decomposition pri
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combinations of known solutions of
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lower bound subproblem (parts 1 and
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&.',* and CI::;, are a 1 x (i - 1)
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from the same aggregated subproblem
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problems is discussed. Various prop
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Processor 2 Step O. Set level I cou
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- update and solve spi ; record opt
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for the optimum by exchanging the p
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The next result guarantees that in
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fonn a nondecreasing senes of lower
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one upper bound subproblem. The alg
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su, , for al1 i Note that when k= 1
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with proposals or cuts from other s
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- if t is an upper bound subproblem
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Proof : The Assumption guarantees t
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nNk-'. Proof : For i= 1.2. ... . N
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Chapter 5 Preliminary Implementatio
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5.1.1 Decomposition Phase In the de
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NB,~,PEI 8 # Selectmg the rows ROWS
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Figure 53 Example of multi-part str
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in each computer as discussed in Ch
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followiag 4 parts: scenarios 1 to 1
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"WSET' is the time taken on the RSI
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esults of the parallel decompositio
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Table 5.6 Performance of Parailel D
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We dso anaiyzed the speed-ups and e
Page 119 and 120: Chapter 6 Conclusions and Future Re
Page 121 and 122: 8. tested several models for conver
Page 123 and 124: computational speed for some real w
Page 125 and 126: APPENDIX A GAMS codes of the test m
Page 127 and 128: hydrA 25, ...., hydrZ 1s /; paramet
Page 129 and 130: NwatBal(hydro,u,pericds,senarios1 .
Page 131 and 132: hydrEI 1100, hydrB 1200, ..... hydr
Page 133 and 134: VARIABLES En 'Expectation of the ut
Page 135 and 136: APPENDIX B The C codes of WSET We p
Page 137 and 138: k denoces an index of SubProblern t
Page 139 and 140: SPb~[kl-+; 1 else ( if (rowinfoIi]
Page 141 and 142: * Pack initial &ta and send '/ for(
Page 143 and 144: Appendiv C C codes of WATPAR We pre
Page 145 and 146: if (LpSub [k] ->rwnmbs [pl > HISPb-
Page 147 and 148: * Use advanced basis at each iterat
Page 149 and 150: status = CPXnewcols (env,lp, 1, &Lp
Page 151 and 152: TERMINATE : ..,. Free up the proble
Page 153 and 154: short SolveProb(C3XEWptr env, CP.XL
Page 155 and 156: variables and add other part's link
Page 157 and 158: i* Add a cut for upper level iterat
Page 159 and 160: C3 Proecssor 3 : Lower-Upper Bound
Page 161 and 162: t for(k=l; k c icer-count; k+-1 sta
Page 163 and 164: { gectimeofday(&tvl, (struct cimezo
Page 165 and 166: i status = CPXchgcoef(env, lp, (int
Page 167 and 168: status = AddLamcols ( LpSub, env, I
Page 169: BIBLIOGRAPHY Aardal. K. And A. An.
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